I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).

Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in $\mathbb{P}^n$ for any $n$ (nb: this is different from asking if every smooth curve in $\mathbb{P}^n$ is a complete intersection, which is of course false; e.g. the twisted cubic)? I expect that the answer is "yes", though it might be "no" for specific genera (and I'd be interested in known these genera). If you bounded $n$, then probably you could use the fact that the moduli space of curves is of general type for large genus to prove this.

Fix a genus $g$. Does there exist some $n$ such that $\mathbb{P}^n$ contains a smooth genus $g$ curve as a complete intersection? I'm not really sure if the answer should be yes or no; if it is no, then I'd be interested in knowing which $g$ satisfy this.